Properties

Label 81.4.c.b.55.1
Level $81$
Weight $4$
Character 81.55
Analytic conductor $4.779$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 81 = 3^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 81.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.77915471046\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 9)
Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

Embedding invariants

Embedding label 55.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 81.55
Dual form 81.4.c.b.28.1

$q$-expansion

\(f(q)\) \(=\) \(q+(4.00000 + 6.92820i) q^{4} +(-10.0000 + 17.3205i) q^{7} +O(q^{10})\) \(q+(4.00000 + 6.92820i) q^{4} +(-10.0000 + 17.3205i) q^{7} +(35.0000 + 60.6218i) q^{13} +(-32.0000 + 55.4256i) q^{16} +56.0000 q^{19} +(62.5000 - 108.253i) q^{25} -160.000 q^{28} +(-154.000 - 266.736i) q^{31} +110.000 q^{37} +(260.000 - 450.333i) q^{43} +(-28.5000 - 49.3634i) q^{49} +(-280.000 + 484.974i) q^{52} +(-91.0000 + 157.617i) q^{61} -512.000 q^{64} +(440.000 + 762.102i) q^{67} +1190.00 q^{73} +(224.000 + 387.979i) q^{76} +(-442.000 + 765.566i) q^{79} -1400.00 q^{91} +(665.000 - 1151.81i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{4} - 20 q^{7} + O(q^{10}) \) \( 2 q + 8 q^{4} - 20 q^{7} + 70 q^{13} - 64 q^{16} + 112 q^{19} + 125 q^{25} - 320 q^{28} - 308 q^{31} + 220 q^{37} + 520 q^{43} - 57 q^{49} - 560 q^{52} - 182 q^{61} - 1024 q^{64} + 880 q^{67} + 2380 q^{73} + 448 q^{76} - 884 q^{79} - 2800 q^{91} + 1330 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/81\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(3\) 0 0
\(4\) 4.00000 + 6.92820i 0.500000 + 0.866025i
\(5\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(6\) 0 0
\(7\) −10.0000 + 17.3205i −0.539949 + 0.935220i 0.458957 + 0.888459i \(0.348223\pi\)
−0.998906 + 0.0467610i \(0.985110\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(12\) 0 0
\(13\) 35.0000 + 60.6218i 0.746712 + 1.29334i 0.949391 + 0.314098i \(0.101702\pi\)
−0.202679 + 0.979245i \(0.564965\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −32.0000 + 55.4256i −0.500000 + 0.866025i
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 56.0000 0.676173 0.338086 0.941115i \(-0.390220\pi\)
0.338086 + 0.941115i \(0.390220\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(24\) 0 0
\(25\) 62.5000 108.253i 0.500000 0.866025i
\(26\) 0 0
\(27\) 0 0
\(28\) −160.000 −1.07990
\(29\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(30\) 0 0
\(31\) −154.000 266.736i −0.892233 1.54539i −0.837192 0.546908i \(-0.815805\pi\)
−0.0550403 0.998484i \(-0.517529\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 110.000 0.488754 0.244377 0.969680i \(-0.421417\pi\)
0.244377 + 0.969680i \(0.421417\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(42\) 0 0
\(43\) 260.000 450.333i 0.922084 1.59710i 0.125900 0.992043i \(-0.459818\pi\)
0.796184 0.605054i \(-0.206849\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(48\) 0 0
\(49\) −28.5000 49.3634i −0.0830904 0.143917i
\(50\) 0 0
\(51\) 0 0
\(52\) −280.000 + 484.974i −0.746712 + 1.29334i
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(60\) 0 0
\(61\) −91.0000 + 157.617i −0.191006 + 0.330832i −0.945584 0.325379i \(-0.894508\pi\)
0.754578 + 0.656210i \(0.227842\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −512.000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 440.000 + 762.102i 0.802307 + 1.38964i 0.918094 + 0.396362i \(0.129728\pi\)
−0.115787 + 0.993274i \(0.536939\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 1190.00 1.90793 0.953966 0.299916i \(-0.0969588\pi\)
0.953966 + 0.299916i \(0.0969588\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 224.000 + 387.979i 0.338086 + 0.585583i
\(77\) 0 0
\(78\) 0 0
\(79\) −442.000 + 765.566i −0.629480 + 1.09029i 0.358177 + 0.933654i \(0.383399\pi\)
−0.987656 + 0.156637i \(0.949935\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) −1400.00 −1.61275
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 665.000 1151.81i 0.696088 1.20566i −0.273725 0.961808i \(-0.588256\pi\)
0.969813 0.243851i \(-0.0784109\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 1000.00 1.00000
\(101\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(102\) 0 0
\(103\) −910.000 1576.17i −0.870534 1.50781i −0.861446 0.507850i \(-0.830440\pi\)
−0.00908799 0.999959i \(-0.502893\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) −646.000 −0.567666 −0.283833 0.958874i \(-0.591606\pi\)
−0.283833 + 0.958874i \(0.591606\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −640.000 1108.51i −0.539949 0.935220i
\(113\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 665.500 + 1152.68i 0.500000 + 0.866025i
\(122\) 0 0
\(123\) 0 0
\(124\) 1232.00 2133.89i 0.892233 1.54539i
\(125\) 0 0
\(126\) 0 0
\(127\) 380.000 0.265508 0.132754 0.991149i \(-0.457618\pi\)
0.132754 + 0.991149i \(0.457618\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(132\) 0 0
\(133\) −560.000 + 969.948i −0.365099 + 0.632370i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(138\) 0 0
\(139\) −1288.00 2230.88i −0.785948 1.36130i −0.928431 0.371504i \(-0.878842\pi\)
0.142484 0.989797i \(-0.454491\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 440.000 + 762.102i 0.244377 + 0.423273i
\(149\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(150\) 0 0
\(151\) −874.000 + 1513.81i −0.471027 + 0.815843i −0.999451 0.0331378i \(-0.989450\pi\)
0.528424 + 0.848981i \(0.322783\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 1925.00 + 3334.20i 0.978546 + 1.69489i 0.667699 + 0.744432i \(0.267279\pi\)
0.310847 + 0.950460i \(0.399387\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −3400.00 −1.63379 −0.816897 0.576783i \(-0.804308\pi\)
−0.816897 + 0.576783i \(0.804308\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(168\) 0 0
\(169\) −1351.50 + 2340.87i −0.615157 + 1.06548i
\(170\) 0 0
\(171\) 0 0
\(172\) 4160.00 1.84417
\(173\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(174\) 0 0
\(175\) 1250.00 + 2165.06i 0.539949 + 0.935220i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 3458.00 1.42006 0.710031 0.704171i \(-0.248681\pi\)
0.710031 + 0.704171i \(0.248681\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(192\) 0 0
\(193\) 575.000 + 995.929i 0.214453 + 0.371443i 0.953103 0.302646i \(-0.0978698\pi\)
−0.738650 + 0.674089i \(0.764536\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 228.000 394.908i 0.0830904 0.143917i
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) −5236.00 −1.86518 −0.932588 0.360942i \(-0.882455\pi\)
−0.932588 + 0.360942i \(0.882455\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) −4480.00 −1.49342
\(209\) 0 0
\(210\) 0 0
\(211\) −3016.00 5223.87i −0.984028 1.70439i −0.646177 0.763188i \(-0.723633\pi\)
−0.337852 0.941199i \(-0.609700\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 6160.00 1.92704
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 1610.00 2788.60i 0.483469 0.837393i −0.516351 0.856377i \(-0.672710\pi\)
0.999820 + 0.0189844i \(0.00604328\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(228\) 0 0
\(229\) −2233.00 3867.67i −0.644370 1.11608i −0.984447 0.175684i \(-0.943786\pi\)
0.340076 0.940398i \(-0.389547\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(240\) 0 0
\(241\) 3689.00 6389.54i 0.986014 1.70783i 0.348673 0.937244i \(-0.386632\pi\)
0.637341 0.770582i \(-0.280034\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −1456.00 −0.382012
\(245\) 0 0
\(246\) 0 0
\(247\) 1960.00 + 3394.82i 0.504906 + 0.874523i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −2048.00 3547.24i −0.500000 0.866025i
\(257\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(258\) 0 0
\(259\) −1100.00 + 1905.26i −0.263902 + 0.457092i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −3520.00 + 6096.82i −0.802307 + 1.38964i
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 812.000 0.182013 0.0910064 0.995850i \(-0.470992\pi\)
0.0910064 + 0.995850i \(0.470992\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 2015.00 3490.08i 0.437074 0.757035i −0.560388 0.828230i \(-0.689348\pi\)
0.997462 + 0.0711951i \(0.0226813\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(282\) 0 0
\(283\) −2800.00 4849.74i −0.588137 1.01868i −0.994476 0.104961i \(-0.966528\pi\)
0.406340 0.913722i \(-0.366805\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −4913.00 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 4760.00 + 8244.56i 0.953966 + 1.65232i
\(293\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 5200.00 + 9006.66i 0.995758 + 1.72470i
\(302\) 0 0
\(303\) 0 0
\(304\) −1792.00 + 3103.84i −0.338086 + 0.585583i
\(305\) 0 0
\(306\) 0 0
\(307\) 10640.0 1.97804 0.989018 0.147797i \(-0.0472182\pi\)
0.989018 + 0.147797i \(0.0472182\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(312\) 0 0
\(313\) −5005.00 + 8668.91i −0.903832 + 1.56548i −0.0813539 + 0.996685i \(0.525924\pi\)
−0.822478 + 0.568797i \(0.807409\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −7072.00 −1.25896
\(317\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 8750.00 1.49342
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −496.000 + 859.097i −0.0823644 + 0.142659i −0.904265 0.426971i \(-0.859580\pi\)
0.821901 + 0.569631i \(0.192914\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 2465.00 + 4269.51i 0.398448 + 0.690133i 0.993535 0.113529i \(-0.0362155\pi\)
−0.595086 + 0.803662i \(0.702882\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −5720.00 −0.900440
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(348\) 0 0
\(349\) 5957.00 10317.8i 0.913670 1.58252i 0.104834 0.994490i \(-0.466569\pi\)
0.808837 0.588033i \(-0.200098\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −3723.00 −0.542790
\(362\) 0 0
\(363\) 0 0
\(364\) −5600.00 9699.48i −0.806373 1.39668i
\(365\) 0 0
\(366\) 0 0
\(367\) −2170.00 + 3758.55i −0.308646 + 0.534591i −0.978066 0.208293i \(-0.933209\pi\)
0.669420 + 0.742884i \(0.266543\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −6175.00 10695.4i −0.857183 1.48469i −0.874605 0.484837i \(-0.838879\pi\)
0.0174213 0.999848i \(-0.494454\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −8584.00 −1.16340 −0.581702 0.813402i \(-0.697613\pi\)
−0.581702 + 0.813402i \(0.697613\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 10640.0 1.39218
\(389\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 1190.00 0.150439 0.0752196 0.997167i \(-0.476034\pi\)
0.0752196 + 0.997167i \(0.476034\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 4000.00 + 6928.20i 0.500000 + 0.866025i
\(401\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(402\) 0 0
\(403\) 10780.0 18671.5i 1.33248 2.30793i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −4123.00 7141.25i −0.498458 0.863354i 0.501541 0.865134i \(-0.332767\pi\)
−0.999998 + 0.00177990i \(0.999433\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 7280.00 12609.3i 0.870534 1.50781i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(420\) 0 0
\(421\) −8569.00 + 14841.9i −0.991989 + 1.71818i −0.386597 + 0.922249i \(0.626350\pi\)
−0.605392 + 0.795927i \(0.706984\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −1820.00 3152.33i −0.206267 0.357265i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) −2590.00 −0.287454 −0.143727 0.989617i \(-0.545909\pi\)
−0.143727 + 0.989617i \(0.545909\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −2584.00 4475.62i −0.283833 0.491613i
\(437\) 0 0
\(438\) 0 0
\(439\) −7462.00 + 12924.6i −0.811257 + 1.40514i 0.100728 + 0.994914i \(0.467883\pi\)
−0.911985 + 0.410224i \(0.865450\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 5120.00 8868.10i 0.539949 0.935220i
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −6355.00 + 11007.2i −0.650491 + 1.12668i 0.332513 + 0.943099i \(0.392103\pi\)
−0.983004 + 0.183585i \(0.941230\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(462\) 0 0
\(463\) 9890.00 + 17130.0i 0.992716 + 1.71943i 0.600698 + 0.799476i \(0.294889\pi\)
0.392017 + 0.919958i \(0.371777\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) −17600.0 −1.73282
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 3500.00 6062.18i 0.338086 0.585583i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(480\) 0 0
\(481\) 3850.00 + 6668.40i 0.364958 + 0.632126i
\(482\) 0 0
\(483\) 0 0
\(484\) −5324.00 + 9221.44i −0.500000 + 0.866025i
\(485\) 0 0
\(486\) 0 0
\(487\) 20900.0 1.94470 0.972351 0.233526i \(-0.0750265\pi\)
0.972351 + 0.233526i \(0.0750265\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 19712.0 1.78447
\(497\) 0 0
\(498\) 0 0
\(499\) 7568.00 + 13108.2i 0.678938 + 1.17596i 0.975301 + 0.220880i \(0.0708930\pi\)
−0.296363 + 0.955075i \(0.595774\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 1520.00 + 2632.72i 0.132754 + 0.229937i
\(509\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(510\) 0 0
\(511\) −11900.0 + 20611.4i −1.03019 + 1.78433i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 0 0
\(523\) −12040.0 −1.00664 −0.503320 0.864100i \(-0.667888\pi\)
−0.503320 + 0.864100i \(0.667888\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 6083.50 10536.9i 0.500000 0.866025i
\(530\) 0 0
\(531\) 0 0
\(532\) −8960.00 −0.730198
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −22678.0 −1.80222 −0.901112 0.433586i \(-0.857248\pi\)
−0.901112 + 0.433586i \(0.857248\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −820.000 + 1420.28i −0.0640963 + 0.111018i −0.896293 0.443463i \(-0.853750\pi\)
0.832196 + 0.554481i \(0.187083\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −8840.00 15311.3i −0.679774 1.17740i
\(554\) 0 0
\(555\) 0 0
\(556\) 10304.0 17847.1i 0.785948 1.36130i
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) 36400.0 2.75413
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(570\) 0 0
\(571\) −11656.0 20188.8i −0.854270 1.47964i −0.877320 0.479905i \(-0.840671\pi\)
0.0230498 0.999734i \(-0.492662\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −17710.0 −1.27778 −0.638888 0.769300i \(-0.720605\pi\)
−0.638888 + 0.769300i \(0.720605\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(588\) 0 0
\(589\) −8624.00 14937.2i −0.603303 1.04495i
\(590\) 0 0
\(591\) 0 0
\(592\) −3520.00 + 6096.82i −0.244377 + 0.423273i
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(600\) 0 0
\(601\) 14651.0 25376.3i 0.994387 1.72233i 0.405567 0.914065i \(-0.367074\pi\)
0.588820 0.808264i \(-0.299593\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −13984.0 −0.942054
\(605\) 0 0
\(606\) 0 0
\(607\) 14210.0 + 24612.4i 0.950191 + 1.64578i 0.745007 + 0.667056i \(0.232446\pi\)
0.205184 + 0.978723i \(0.434221\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 17390.0 1.14580 0.572900 0.819625i \(-0.305818\pi\)
0.572900 + 0.819625i \(0.305818\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(618\) 0 0
\(619\) 13328.0 23084.8i 0.865424 1.49896i −0.00120126 0.999999i \(-0.500382\pi\)
0.866625 0.498959i \(-0.166284\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −7812.50 13531.6i −0.500000 0.866025i
\(626\) 0 0
\(627\) 0 0
\(628\) −15400.0 + 26673.6i −0.978546 + 1.69489i
\(629\) 0 0
\(630\) 0 0
\(631\) 1892.00 0.119365 0.0596825 0.998217i \(-0.480991\pi\)
0.0596825 + 0.998217i \(0.480991\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 1995.00 3455.44i 0.124089 0.214929i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(642\) 0 0
\(643\) −6580.00 11396.9i −0.403561 0.698989i 0.590592 0.806971i \(-0.298894\pi\)
−0.994153 + 0.107982i \(0.965561\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −13600.0 23555.9i −0.816897 1.41491i
\(653\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(660\) 0 0
\(661\) 10241.0 + 17737.9i 0.602615 + 1.04376i 0.992423 + 0.122864i \(0.0392080\pi\)
−0.389808 + 0.920896i \(0.627459\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −12025.0 + 20827.9i −0.688751 + 1.19295i 0.283491 + 0.958975i \(0.408508\pi\)
−0.972242 + 0.233977i \(0.924826\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −21624.0 −1.23031
\(677\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(678\) 0 0
\(679\) 13300.0 + 23036.3i 0.751704 + 1.30199i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 16640.0 + 28821.3i 0.922084 + 1.59710i
\(689\) 0 0
\(690\) 0 0
\(691\) 8036.00 13918.8i 0.442408 0.766273i −0.555460 0.831543i \(-0.687458\pi\)
0.997868 + 0.0652705i \(0.0207910\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −10000.0 + 17320.5i −0.539949 + 0.935220i
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 6160.00 0.330482
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −18073.0 + 31303.4i −0.957328 + 1.65814i −0.228381 + 0.973572i \(0.573343\pi\)
−0.728948 + 0.684569i \(0.759990\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 36400.0 1.88018
\(722\) 0 0
\(723\) 0 0
\(724\) 13832.0 + 23957.7i 0.710031 + 1.22981i
\(725\) 0 0
\(726\) 0 0
\(727\) 5390.00 9335.75i 0.274971 0.476264i −0.695157 0.718858i \(-0.744665\pi\)
0.970128 + 0.242594i \(0.0779984\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −7525.00 13033.7i −0.379184 0.656767i 0.611759 0.791044i \(-0.290462\pi\)
−0.990944 + 0.134277i \(0.957129\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 31376.0 1.56182 0.780910 0.624644i \(-0.214756\pi\)
0.780910 + 0.624644i \(0.214756\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 11726.0 + 20310.0i 0.569757 + 0.986849i 0.996590 + 0.0825179i \(0.0262962\pi\)
−0.426832 + 0.904331i \(0.640370\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −41470.0 −1.99109 −0.995543 0.0943039i \(-0.969937\pi\)
−0.995543 + 0.0943039i \(0.969937\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(762\) 0 0
\(763\) 6460.00 11189.0i 0.306511 0.530892i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 2303.00 + 3988.91i 0.107995 + 0.187053i 0.914958 0.403549i \(-0.132224\pi\)
−0.806963 + 0.590602i \(0.798890\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −4600.00 + 7967.43i −0.214453 + 0.371443i
\(773\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 0 0
\(775\) −38500.0 −1.78447
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 3648.00 0.166181
\(785\) 0 0
\(786\) 0 0
\(787\) −21700.0 37585.5i −0.982874 1.70239i −0.651029 0.759053i \(-0.725662\pi\)
−0.331844 0.943334i \(-0.607671\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −12740.0 −0.570505
\(794\) 0 0
\(795\) 0 0
\(796\) −20944.0 36276.1i −0.932588 1.61529i
\(797\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0